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3D Coordinate Geometry

Append a z-axis to the 2-dimensional plane and conquer the realm of 3-dimensional space.

Level 2

         

Point \(P\) is some point on the surface of the sphere \[ {(x-1)}^{2}+{(y+2)}^{2}+{(z-3)}^{2}=1 .\] What is the shortest possible distance between \(P\) and \(O=(0, 0, 0)?\)

If the two lines \[\begin{align} \frac {x-1 }{k } &=\frac { y+1 }{2 } =z,\\ \\ \\ \frac {x+2 }{-3 } & =1-y =\frac{z+2}{k} \end{align}\] are perpendicular to each other, then what is the value of \(k?\)

The points \(A\), \(B\), \(P\), and \(Q\) all lie on one line, with \(A=(-3, 5, 8).\)

Point \(P\) lies in the \(xy\)-plane between \(A\) and \(B\) such that the distance from \(A\) to \(P\) is twice the distance from \(P\) to \(B\). Furthermore, \(Q\) sits on the \(z\)-axis such that the distance from \(A\) to \(Q\) is twice the distance from \(Q\) to \(B\).

If \(B=(x,y,z),\) what is the value of \(x+y+z?\)

An infinite column is centered along the \(z\)-axis. It has a square cross section of side length equal to 10. It is cut by the plane \(4x - 7y + 4z = 25.\). What is the area of the surface cut?

If the point \(Q(a, b, c)\) is the reflection of the point \(P(-6, 2, 3)\) about the plane \(3x-4y+5z-9=0\). Determine the value of \(a+b+c\).

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